# 3D slope

I know that slope in 2D = $$\frac{\Delta{y}}{\Delta{x}}$$.

I mean for every line in 3D there are 3 2D slopes. Those are:
$$\frac{\Delta{y}}{\Delta{x}}$$ $$\frac{\Delta{y}}{\Delta {z}}$$ and $$\frac{\Delta{z}}{\Delta{x}}$$

But how do you combine those 3 slopes to form 1 3D slope?

And if you think this is homework it isn't. I am just trying to extrapolate the concept of the slope of a line to lines in higher dimensions.

There are an infinite amount of directions to have slope on a 3-D graph. You may want to look into partial derivatives (which are generally confined to 2 different directions). However, using what's called directional derivatives, you can find the slope in any direction.

But for a single line instead of a plane there is 1 and only 1 direction for the slope because of the line being described in 3D instead of 2D. This means that there should be a way to combine the 3 2D slopes into 1 3D slope.

SteamKing
Staff Emeritus
Homework Helper
But for a single line instead of a plane there is 1 and only 1 direction for the slope because of the line being described in 3D instead of 2D. This means that there should be a way to combine the 3 2D slopes into 1 3D slope.
You might want to express your 3-D line parametrically or use a vector representation.

http://mathworld.wolfram.com/Line.html

While the slope of a line in 2-dimensions is a handy thing to know, things get more complicated in 3-dimensions, which is why different formulations for the equation of the line in space have been adopted.